3 ms·
When you randomize the event occurences, you create intervals that are shorter and longer than average, so that a random point is more likely to be in a longer
by yorwba 2y ago
When you randomize the event occurences, you create intervals that are shorter and longer than average, so that a random point is more likely to be in a longer interval, so that the expected length of the interval containing a random point is greater than the expected length of a random interval.
To see this, consider just two intervals of length x and 2-x, i.e. 1 on average. A random point is in the first interval x/2 of the time and in the second one the other 1-x/2 of the time, so the expected length of the interval containing a random point is x/2 * x + (1-x/2) * (2-x) = x² - 2x + 2, which is 1 for x = 1 but larger everywhere else, reaching 2 for x = 0 or 2.
- fc417fc802 2y agoI think I understand my mistake. As the variance of the intervals widens the average event interval remains the same but the expected average distances for a sample point change. (For some reason I thought that average distances wouldn't change. I'm not sure why.) Your example illustrates it nicely. A more intuitive way of illustrating the math might be to suppose 1 event per 10 minutes but they always happen in pairs simultaneously (20 minute gap), or in triplets simultaneously (30 minute gap), or etc. So effectively the earlier example that I replied to is the birthday paradox, with N people, sampling a day at random, and asking how far from a birthday you expect to be on either side. If that counts as a paradox then so does the number of upvotes my reply received.